The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Sophie Germain

As many would know– or should know– successful females throughout history have either been scorned, ignored, or referred to briefly from a footnote. As discouraging as that is, many future generations have used that as motivation to make greater accomplishments in areas that traditionally were not open to them. Why, Elizabeth I made England one of the most powerful nations in Europe, Marie Curie discovered Radium and Polonium, and the great Marie-Sophie Germain pioneered elasticity theory.

Wait, who? Elasti-what? Why, everyone knows how Germain was an independent woman who didn’t need a career or formal education to show her what’s what. Obviously. I mean, she only built the foundations for Fermat’s Last Theorem (FLT)– which I’m sure you’re familiar with– and used her own method to show that the modular arithmetic conditions on the exponent n from FLT can lead to particular conditions for x, y, and z for the equation xn + yn = zn. (FLT states that no positive integers x, y, and z can satisfy the previous equation for any integer n greater than 2.)

Fermat’s Last Theorem. Image: Phuong Trinh, via Wikimedia Commons.

Germain had intended on proving many cases of FLT all at once rather than working through each one individually. Of course, this was not a very successful attempt, but her work helped support future generations with their studies on FLT.

Germain’s journey into the world of number theory started with a letter to polynomials and transformation master Adrien-Marie Legendre, when his work Essai sur la théorie des nombres (Essay on Number Theory) was published. In short, they became pen pals and corresponded with each other for many years regarding number theory and eventually elasticity. Legendre was even generous enough to add Germain to a magnificent and grand footnote in his treatise on number theory. And why would he do such a thing? Well, Germain’s unpublished manuscript called Remarque sur l’impossibilité de satisfaire en nombres entiers a l’équation xp + yp = zp, which argued that counterexamples for FLT for p > 5 would need to be numbers nearly 40 digits long, was so brilliant that Legendre used it to prove FLT for p = 5. Afterwards, Germain sent a paper on analysis to the magnificent J. L. Lagrange (one of the big daddies of analysis, number theory, and classical and celestial mechanics) as “Monsieur Antoine-Auguste Le Blanc” and impressed him so much that he became a mentor and supporter of her work, even after Germain confessed to him that she was a woman and it was she who was using the name of a former male student of his to correspond with him. This was also the same with Carl Friedrich Gauss– math guru in number theory, statistics, algebra, matrix theory, differential geometry, optics, analysis, electrostatistics, geodesy, astronomy, and geophysics (phew!). She wrote to him using “Le Blanc” again to discuss number theory, which she studied thoroughly in Gauss’s Disquisitiones Arithmeticae and offered her own work on the theorems listed. Despite her work not having the proper structure that normally would have been apparent from a formal education, Gauss had used Germain’s ideas and proofs for FLT. One of Germain’s unsupported proofs was for the case n = p, with p being a prime number with the form p = 8k + 7. Gauss would provide counterexamples to some of Germain’s proofs as the years went on. Their correspondence would later end due to Gauss no longer expressing any interest in number theory. He moved on to other mathematical fields. If was from there that Germain went on her own tangent.

Caption: Germain’s Elasticity Theorem submitted for the third and final time for the contest. Image: Rational Wiki

Starting in 1809, she began her work on elasticity, specifically with the theory of vibrating elastic surfaces using vibrating metal plates. The Paris Academy of Sciences was having a contest to elaborate E. F. F. Chladni’s study on the subject, and Germain was the only contestant. Well, thanks to the lack of formal education she did not receive the prize due to unsupported work. But the judges thought her results were impressive, so you could say she got the equivalent of a participation sticker. If there’s ever a time to be grateful for a teenage rebellion that resulted in a lifelong pen pal, it was right and there. Lagrange had helped correct Germain’s mistakes, and before long she entered the contest again in 1813. Oh but wait, that whole informal education worked its magic again and she didn’t win again. But this time, she had an honorable mention only because her work had taken an approach that wasn’t derived from physics. And also because there were still several mistakes in her calculations. But hey, third time’s the charm right? The year of 1816 was Germain’s time to shine when she finally won and had her work once again criticized for not completely resulting in what was expected. Ah, c’est la vie. Although her work was not fully supported, it would later become the critical stepping stone for future generations who aimed to significantly improve her work. Now all that was left was to be accepted into society as an educated woman to extend on the topic she had been working on for the past 16 years and become a renowned mathematician for years to come.

Yeah. It’d still take a century or two for that to happen. So why aren’t we teaching our kids about Miss Marie-Sophie Germain or any of her work in schools? It’s only number theory and advanced mathematics. Germain started in 1789 when she was 13 and studied differential calculus so much that her parents found her incurable of her newfound disease. She went through her teenage rebellion in 1794 at the age of 18 when she began to make friends with students at the male-only Ecole Polytechnique and took their lecture notes to study. Did I mention that the majority of Europe at the time did not accept women into colleges? The only exceptions were the wealthy upper class women, only so that they could have more ice breakers when gathering at social functions. Germain, however, was only a middle class maiden who was pushed from exploring her interests. She continued to do what she could outside of the education system. As time went on, Germain continued to be excluded from any sort of research related to mathematics. Any work she did submit to educational institutions were not treated “as a man’s”. Germain’s essays weren’t formally rejected, which meant they actually were rejected but in a very rude way. The logic at the time for that was if the institution sent a letter of rejection to her, they would technically be acknowledging a woman’s work, thus making her work equivalent to a male’s. They didn’t want to portray that to Germain or to anyone, because– you know– that just wasn’t acceptable in the 18th century. This didn’t deter her though. She continued for the rest of her life working on elasticity and math theory, along with philosophy and psychology.

Germain would later die from breast cancer, after submitting a paper to Crelle’s Journal in 1831 explaining elastic surfaces and their curvature. Though this seems anything but the happy ending Germain expected, her life’s work greatly improved the world of math. And not only that, her presence in education during the 18th century demonstrated that the difficulties women have in pursuing math and science were meant to be respected, not ostracized. Gauss became one of the supporters for women’s social justice after he had found out Germain was a woman. He recommended Germain to receive an honorary degree in mathematics before her death, and had exposed the unfair treatment European women faced in education and in social settings.

After our discussion in class about the work of Sophie Germain, I was interested in learning more about other prominent women in mathematics. I’m sure we will go over some of them in class, but here is what I discovered about some very smart women.

One of the earliest known female mathematicians was Hypatia. She lived in the time period of approximately 350-416 C.E. She was excellent at mathematics, astronomy and philosophy. No doubt this is because her father was Theon, one of the last members of the library of Alexandria. Unfortunately for us, we do not know many of her contributions to science. She is more well known for her brutal death. She was riding in her carriage, when she was forcefully removed, stripped, beaten to death, and then her body was burned. Not a nice way to go. Regardless, of that cruelty, she is one of the first well known women mathematicians, and in her time that was quite an accomplishment.

Another leading lady in mathematics was Ada Lovelace. She lived from 1815-1852 as the daughter of well known writer, Lord Byron. She never met her father, and her mother advocated her to study fields that were different from language and poems. Essentially, anything different from what her father was well known for. It must have been a bad break up. Thus, math and science it was. Turns out, she is credited with being the world’s first programmer. But before that achievement, she demonstrated ingenuity as a child. She set her mind toward the daunting task of flying, at the young age of twelve. She researched materials, how to build wings, and even wanted to incorporated steam! Being curious from a young age really inspired her to continue her study of the sciences.

Because of the strict laws against the education of women she had to study mathematics with a tutor, she could not technically enroll in university. She met Charles Babbage later in life and their friendship encouraged her studies. They continued their correspondence even after her marriage to the Earl of Lovelace. At the time Babbage was working on a theoretical machine called the Analytical Engine. The idea was that the Engine could store numbers, and it could do long cycles and loops without the help of people. She wrote to Babbage about including Bernoulli numbers and how such implicit functions could be solved by the Engine. According to Wolfram Alpha, “The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function. These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.” In order to calculate Bernoulli numbers, there must be a lot of operations involved. To top it off, they anticipated that the Analytical Engine could perform this task. Below I have pictured one of Ada’s tables on how she envisioned the Engine could compute this. Remarkably enough, Lady Lovelace once said, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with.” She understood that the machine is only as good as the people who are using it. It cannot come up with new ideas, or understand why it is doing computation, it can only do said computation. If this machine were to have been made, it would have been an incredible invention. However, the fact that it was never brought to production, does not in any way reduce all of the work both Ada and Charles did.

Unfortunately, there has been speculation that Ada did not contribute in the mathematical sense, but was merely a notetaker for Babbage. This is baffling because in his autobiography, Babbage gives her credit for all of the theoretical math she did for his Analytical Engine. I could continue this post with a commentary about women in science even today, but I’d better move onto the final female mathematician I wish to recognize.

The final female mathematician I wish to discuss is Emmy Noether. Emmy was born in Germany in the late 1800’s. She was denied a lot of formal education because she was a woman. She began her studies with piano and languages, but soon discovered a passion for math, like her father, and her brother. Universities in Germany were hesitant to let her become a professor, although, she did get the status of Associate Professor eventually. This title was taken away however, when the Nazi’s came to power because she was Jewish. Despite all of this, she had many notable accomplishments. So much so, that Albert Einstein once referred to her as “the most significant creative mathematical genius thus far produced since the higher education of women began.” This is high praise, especially coming from a man our society reveres as the most intelligent man ever known.

She was behind a revolutionary theorem, called Noether’s Theorem. This theorem states that: “Each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to conservation of momentum, symmetry under rotation to conservation of angular momentum, symmetry in time to conservation of energy, etc.” And when I first read this, I was quite confused. However, with some help from my sources, I was able to wrap my mind around it to a certain extent. Noether is telling us that when we find symmetrical things, in nature or otherwise, there is some sort of conservation force that goes with it. One example of this, that is referenced in the New York Times article, is the relationship between time and energy. To paraphrase, if a person throws a ball up in the air right now, or throws it the same way sometime in the future, the time does not affect the trajectory of the ball. This means that the symmetry of time is related to the conservation of energy. This is crucial to how we think about physics today, and I could definitely relate this to my old physics teacher being like a broken record and telling us energy cannot be created or destroyed, it only changes form. Emmy clearly made an impact on not only math, but the way we think about certain concepts today. She even developed some of the mathematical formulas that Einstein used for his Theory of Relativity.

It seems to me that Emmy deserves much more recognition than she is receiving. Truthfully, I had not even heard of her until I began research for this blog post. I know this is not a class about how our society can improve, but one way would be to get more women in math and science. It is interesting to think about how limited women once were. I am optimistic about the progress we have made in that regard, but just think about how much further along we could possibly be in terms of figuring out the mysteries of the world if we had help from every person, from every demographic, and every gender. I do not know if this is possible, but inclusion is a nice thought. These ladies kicked butt in their time, and I hope that the women of the present and the future follow their example and continue to do the same.

I have recently learned that October 14th was Ada Lovelace Day! Ada Lovelace Day celebrates women in all areas of science. And because of that, I would like to dedicate this post to all the amazing ladies out there making leaps and bounds in the sciences. You are an inspiration to me, but all young women of the world.

As a female math student, I often find myself in the minority in my classes. In fact, in one class, I am one of only three girls. Now, if I were young and single, I might really appreciate these odds, however, as a mother of four daughters, I find it rather concerning. Recently in class, we studied the work of Sophie Germain, her contributions to mathematics, and the study of Fermat’s Last Theorem. We also discussed some of the challenges she faced as a woman scholar in the early 1800’s. I have been thinking about the roles of women in math, and I have wondered–even with so many programs to encourage women in the fields of math and science, why do we still see such a large gender gap?

Women have truly struggled over the years to have equal opportunities for education, and, while in many subjects women have equal footing, they have been slower to catch up in math and science. Historically, women who have contributed to math have had a difficult time pursuing higher education. Sophie Germain had to use the pseudonym of a male student to submit papers to the university. Almost 60 years later, another promising mathematician, Christine Ladd, also had a difficult time obtaining a fellowship at John Hopkins University because of her gender. Her experience is described as follows:

[T]he university first announced its fellowship program in 1876, and one of the first applications to arrive was one signed “C. Ladd.” The credentials accompanying the application indicated such outstanding ability that a fellowship in mathematics was awarded to the applicant, site [sic] unseen, and was accepted. When it was discovered that the “C.” stood for Christine, several embarrassed trustees argued she had used trickery to gain admission, and the board immediately moved to revoke the offer. They failed to reckon, however, with the irascible Professor James J. Sylvester, stellar member of the first faculty. In 1870 Sylvester had been named the world’s greatest living mathematician by the Encyclopedia Britannica, and his presence at Hopkins was a real coup for the struggling university. He was indispensable and knew it, in an ideal position to insist on virtually anything he wanted; in this case, he had read Christine Ladd’s articles in English mathematical journals, and he insisted upon receiving the obviously gifted young woman as his student. Miss Ladd was admitted as a full-time graduate student in the fall of 1878. Though she held a fellowship for three years, the trustees forbade that her name be printed in circulars with those of other fellows, for fear of setting a precedent. Dissension over her continued presence forced one of the original trustees to resign (Riddle, 2014).

Despite such difficulties, woman continued to push forward in mathematics. Now, women don’t face the same discrimination in education; in fact, there are many programs designed to encourage girls to pursue their interests in math, and research shows that the gender gap may be narrowing in education. An article from Time Magazine, titled “The Myth about the Math Gender Gap”, reports about a study by researchers at University of Wisconsin and University of California, Berkeley. The study found that there was very little difference between the scores of girls and boys on federally mandated tests. They also found that equal numbers of boys and girls were taking advanced math classes in high school. This, researchers concluded, is why we are seeing a decrease in the gender gap. An earlier study from 1990 found that although test scores were equal in the elementary years, boys outpaced the girls in high school years. This coincides with the fact that fewer girls at the time were continuing in higher level math classes (Park, 2008).

However, there does continue to be a gap in higher education. Although equal numbers of male and female students are graduating with bachelor degrees in math, fewer women continue to the graduate level and even fewer to the associate professor level. There are several theories about what may be responsible for this gap. Another article in Time titled, “Explaining the Complicated Women + Math Formula”, explores different thoughts on this subject. The article specifically looks at four different theories—ability, prejudice, interest or choice, and the affect of family roles (Luscombe, 2010).

Ability
“As far as ability goes, studies are pretty clear that on average women and men are about the same at math” (Ibid). While an equal number of male and female students graduate with bachelor degrees in math related fields, only 25% of PhDs are given to females and “at the next level, tenure-track associate professor, the proportion of females shrinks to single digits” (Ibid). With this information, we can conclude that a large number of women who have the ability to continue at the higher level of math related fields are not continuing.

Prejudice
Ceci explored the possibility that women were experiencing prejudice from male college faculty in the hiring process, however, this actually showed to be the opposite. “We found that if candidates of matched ability are applying for a position, women are slightly more likely to get the job,” says Ceci.

Interest and Choice
Williams and Ceci found that the interests of girls may be another factor. While some boys may be very good at math, the girls who excel at math also excel at other subjects. It has also been found that girls tend to be more interested in working with living things rather than inanimate objects. Therefore while capable of excelling at math careers, girls may be choosing to follow other interests.

Family and Social Roles
The final factor they found was that, when given the choice, women would choose to follow family interests over their careers. The typical track to a tenure position came at a time when many women were interested in starting families and opportunities were no longer available if they wanted to come back to their career at a later time. “Ceci and Williams believe that the solution lies in changing the way tenure is attained. ‘The tenure structure in academe demands that women having children make their greatest intellectual contributions contemporaneously with their greatest physical and emotional achievements,’ they write, ‘a feat not expected of men.’ The process could be spaced out, candidates given more time. ‘There’s no reason to do it the way we do it, except tradition,’ says Williams (Luscombe, 2010).”

Although women are making huge strides in narrowing the gender gap in mathematics and I have a lot of reasons to be optimistic that my daughters will have opportunities that women throughout history did not, there are definitely still issues to be addressed. Some feel that women are socialized away from math and science, which may be true, but there are definitely other issues at play. The important thing to recognize as we look to the future is that there are no simple solutions and no one single factor at play. Ultimately, we want to ensure that those women who want to pursue careers in math have every opportunity to do so and that the field is free of the prejudice and stereotypes of the past.

As I have gone through the process of gaining a higher education in mathematics, I have made a startling realization that I am alone. Sure, there are other women in my math classes, but the majority of the students are men. I have had to rely on my own strength and diligence to get through the challenging courses. When I started working on my degree, I had many counselors and professors that discouraged me from entering into such a field due to the fact that it was challenging, and the odds were I would not succeed. Whether this opinion was developed from me being a female or not, I have a hard time believing that a male would receive that same type of consolation. Also, it is a popular belief in our culture here in Utah that most women should not enter into the fields of science and mathematics, and are better off obtaining degrees that will benefit them as homemakers. Hence, most women do not pursue a degree in mathematics or science. It troubled me to know that there are no women that I could turn to for help in my field of choice. In my History of Mathematics class, we have been learning about the great minds of mathematics which have mostly been men. However, last week in class, we learned about Sophie Germain, a woman mathematician. This got me thinking that I have never before heard, or learned about other women in the field of mathematics. I’ve been asking myself, why don’t I know more about these women? So I decided to do some research and find other women who have contributed to the field of mathematics and made it possible for other women, like myself, to gain a higher education.

One of the first known female mathematicians was Hypatia (370-415 A.D.). Her father was a well-educated man, and Hypatia spent a lot of time in the world of education learning from her father. From her father’s teachings, Hypatia become very educated in math, science, and astronomy and would impart this knowledge to students in her home. Large crowds would also come and listen to her teach in the streets. Her fame and popularity, however, turned to be her downfall as she was killed by Christian zealots.

Sophie Germain (1776-1831) was born in a time of revolution, which was shown in her character. During this time, it wasn’t socially acceptable for women to have access to the same education as men. This didn’t stop Sophie from becoming a great mathematician, and being the first woman to win a prize from the French Academy of Sciences for her work on the theory of elasticity. It should be noted that during her life she often worked under a false name to avoid persecution for breaking social boundaries of women in education.

Sofia Kovalevskaya (1850-1891) was born in Russia, where women were not allowed to attend universities. In order for her to pursue some type of higher education, she decided to get married so she could travel to Germany, and was able to be privately tutored by a professor. Sofia was granted a PhD, and went on to produce wonderful works in the fields of mathematics and science, but was always faced with adversity. Despite her hardships, her contributions were vast, and she expanded the opportunities for women in education and women’s rights.

Emmy Noether (1882-1935) grew up in Germany, where she wasn’t allowed to receive a university education. Growing up, she was educated in language, and the common tasks expected from women. At age eighteen, she decided to take courses in mathematics, and was able to become a university student. She received a PhD, and became an unofficial associate professor at the University of Göttingen. However, in 1933 she lost that title because she was Jewish. She decided to move to America and became a lecturer and researcher. There she developed many of the mathematical foundations for Einstein’s general theory of relativity. Einstein later wrote of her that she was “the most significant creative mathematical genius thus far produced since the higher education of women began”(Zielinski).

Ingrid Daubechies (1954-Present) is the first female president of the International Mathematical Union, and is a strong advocate for women in science and mathematics. As a girl, she studied physics and eventually received her PhD, along with other awards. Her most important discovery was in the field of wavelets, which are “mathematical functions useful in digital signal processing and image compression as well as in many other branches of applied and pure mathematics”(Riddle). In a recent interview Daubechies was asked why there is the assumption that men are better at mathematics than women. Her response to this question, “I disagree with this view – completely. There is a highly variable percentage of women in academia and in departments of mathematics across Europe. Differences are so enormous that it becomes obvious that it has something to do with cultural habits, which differ from one nation to another, and not with intelligence”(TWAS).

In conclusion, there have been many women who have made significant contributions to the fields of math and science, and have influenced the works of other male scholars. As a woman in higher education and mathematics, I admire the hardships and work these women accomplished, and wish that more was said about them. In doing this research, I’ve realized I am not alone, and I have many great examples of women who have worked hard and overcame societal obstacles. As a future teacher, I aspire to influence more girls to pursue college degrees and not be intimidated by the “male dominated” subjects, and realize that women are just as intelligent as men.